15 Aspects logiques de la théorie transformationnelleThe class of commutative GIS has the same complexity as the class of commutative groups: the theory of each is decidable. This contrasts strongly with the class of finite (non commutative) GIS, which is not even recursively axiomatizable[Kolman 2004]GIS can be considered as models of a theory in first-order languageThe application of logic that I believe is potentially interesting for finite GIS theory concerns the minimal axiomatic strength of formal systems […] necessary to prove certain finite combinatorial principles of GIS theory. How strong must an axiomatic system be in order to answer (i.e. prove or refute) all the combinatorial problems of finite GIS theory? Can every combinatorial question of finite GIS theory be answered using just finitistic methods? How much mathematics does one need to do music theory?[Kolman 1999]Reverse Mathematical Music Theory?

16 Limitations de la théorie des GIS (Rahn, Tymoczko, …)GIS = (S, G, int)S = ensemble(G,) = groupe d’intervallesint = fonction intervalliquesutintGS  S1. Pour tout objets s, t, u dans S :int(s,t)int(t,u) = int(s,u)?2. Pour tout objet s dans S et tout intervalle i dans G il y a un seul objet t dans S tel que int(s,t) = i« […] Other problems associated with Lewin’s approach to musical intervals -problems that arise even when we can transport arrows from point to point. These result from Lewin’s requirement that intervals always be defined at every point in the space, and that they be represented by functions whose identity is entirely determined by their inputs and outputs. As we will see, these requirements further constrain the range of applicability of Lewin’s theory by prohibiting spaces with boundaries and by eliding the distinct paths that might connect the same pairs of points.[Tymoczko Preprint]

17 Aspects logiques de la théorie transformationnelleThe class of commutative GIS has the same complexity as the class of commutative groups: the theory of each is decidable. This contrasts strongly with the class of finite (non commutative) GIS, which is not even recursively axiomatizableGIS can be considered as models of a theory in first-order language[Kolman 2004]The application of logic that I believe is potentially interesting for finite GIS theory concerns the minimal axiomatic strength of formal systems […] necessary to prove certain finite combinatorial principles of GIS theory. How strong must an axiomatic system be in order to answer (i.e. prove or refute) all the combinatorial problems of finite GIS theory? Can every combinatorial question of finite GIS theory be answered using just finitistic methods? How much mathematics does one need to do music theory?[Reverse Mathematical Music Theory -> ]Trouver de problèmes « finitistically unprovable » qui peuvent être exprimé comme des énoncé [statements] dans la théorie des GIS finis[Kolman 1999]

24 Premières implications philosophiques de l’équivalenceGIS = (S, G, int)Action simplément transitiveCartésianisme vs anti-cartésianisme« To some extent for cultural-historical reasons, it is easier for us to hear “intervals” between individual objects than to hear transpositional relations between them; we are more used to conceiving transpositions as affecting Gestalts built up from individual objects. As this way of talking suggests, we are very much under the influence of Cartesian thinking in such matters. We tend to conceive the primary objects in our musical spaces as atomic individual “elements” rather than contextually articulated phenomena like sets, musical series, and the like. And we tend to imagine ourselves in the position of observers when we theorize about musical space; the space is “out there,” away from our dancing bodies or singing voices. “The interval from s to t” is thereby conceived as modeling a relation of extension, observed in that space external to ourselves; we “see” it out there just as we see distances between holes in a flute or points along a stretched string » [GMIT, p. 158–59].Ti(s)is

25 Premières implications philosophiques de l’équivalenceGIS = (S, G, int)Action simplément transitiveCartésianisme vs anti-cartésianisme« In contrast, the transformational attitude is much less Cartesian. Given locations s and t in our space, this attitude does not ask for some observed measure of extension between reified ‘points’; rather it asks: ‘If I am at s and wish to get to t, what characteristic gesture (e.g. member of STRANS) should I perform in order to arrive there?’ The question generalizes in several important respects: ‘If I want to change Gestalt 1 into Gestalt 2 (as regards to content, or location, or anything else), what sorts of admissible transformations in my space (members of STRANS or otherwise) will do the best job?’ Perhaps none will work completely, but ‘if only ,’ etc. This attitude is by and large the attitude of someone inside the music, as idealized dancer and/or singer. No external observer (analyst/listener) is needed » [GMIT, p. 158–59].Ti(s)is« …instead of regarding the i-arrow […] as a measurement of extension between points s and t observed passively « out there » in a Cartesian res extensa, one can regard the situation actively, like a singer, player or composer, thinking « I am at s; what characteristic transformation do I perform in order to arrive at t? » [GMIT, p. xiii].

26 Premières implications philosophiques de l’équivalenceGIS = (S, G, int)Action simplément transitiveCartésianisme vs anti-cartésianisme« [A]bove we sketched a mathematical dichotomy between intervals in a GIS and transposition-operations on a space: Either can be generated formally from the characteristic properties of the other. More significant than this dichotomy, I believe, is the generalizing power of the transformational attitude. It enables us to subsume the theory of GIS structure, along with the theory of simply transitive groups, into a broader theory of transformations. This enables us to consider intervals-between-things and transpositional-relations-between-Gestalts not as alternatives, but as the same phenomenon manifested in different ways » [GMIT, p. 159].Ti(s)is

34 « Making and Using a Pcset Network for Stockhausen's Klavierstück III »Trois interprétations :HenckKontarskyTudor

35 « Making and Using a Pcset Network for Stockhausen's Klavierstück III »???« The most ‘theoretical’ of the four essays, it focuses on the forms of one pentachord reasonably ubiquitous in the piece. A special group of transformations is developed, one suggested by the musical interrelations of the pentachord forms. Using that group, the essay arranges all pentachord forms of the music into a spatial configuration that illustrates network structure, for this particular phenomenon, over the entire piece. »David Lewin, Musical Form and Transformation, YUP 1993

36 « Making and Using a Pcset Network for Stockhausen's Klavierstück III »Lewin 1993

38 Progression transformationnelle vs réseau transformationnel« Rather than asserting a network that follows pentachord relations one at a time, according to the chronology of the piece, I shall assert instead a network that displays all the pentachord forms used and all their potentially functional interrelationships, in a very compactly organized little spatial configuration. »

39 Reseau transformationnelStockhausen: Klavierstück III (Analyse de D. Lewin)« […] the sequence of events moves within a clearly defined world of possible relationships, and because - in so moving - it makes the abstract space of such a world accessible to our sensibilities. That is to say that the story projects what one would traditionally call form. »

42 Exercices d’écoute Stockhausen: Klavierstück III (Analyse de D. Lewin)« I doubt that [Nicholas ]Cook [cf. A Guide to Music Analysis, 1987] would have much patience with my network analysis; I suspect he would read it as yet one more exercise in what he calls “cracking the code.” Let me be the first to say emphatically that the network analysis is very far from an analysis of the piece, that I find it problematical, and that it took some effort for me to develop the aural agenda of [the ear-training exercises in] example 2.7. »

43 Exercices d’écoute Stockhausen: Klavierstück III (Analyse de D. Lewin)« However, I must say that I enjoyed developing that agenda, which of course I did gradually as my work developed, and not in so neatly packaged a way as in this essay. I felt I was getting at something in the piece that very much involved “what the music did to me,” if only in one of its aspects. I felt I was responding in some measure to a strong sense of challenge I felt about the piece. No matter to what degree I am deluding myself, I miss in Cook the sense of having to extend my ear in response to a sense of challenge ». OpenMusic

48 Progression transformationnelle vs réseau transformationnel« A rational reconstruction of a work or works, which is a theory of the work or works, is an explanation not, assuredly, of the ‘actual’ process of construction, but of how the work or works may be construed by a hearer, how the ‘given’ may be ‘taken’ »M. Babbitt : « Contemporary Music Composition and Music Theory as Contemporary Intellectual History », 1972

50 Autres implications philosophiques de l’approche transformationnelle1. Les groupes comme structures soujacentes des GISThe nature of a given geometry is […] defined by the reference to a determinate group and the way in which spatial forms are related within that type of geometry. [Cf. Felix Klein Erlangen Program ][…] We may raise the question whether there are any concepts and principles that are, although in different ways and different degrees of distinctness, necessary conditions for both the constitution of the perceptual world and the construction of the universe of geometrical thought. It seems to me that the concept of group and the concept of invariance are such principles.E. Cassirer, “The concept of group and the theory of perception”, 1944Felix KleinErnst Cassirer

63 Autres implications philosophiques de l’approche transformationnelle4. Dépassement du cadre positivistico-logique« Because a compelling interpretation of musical perceptions is needed to turn a theoretically true statement into a meaningful statement, analytical judgment plays a central role in meaningfully reducing the scope of the theory. So although it may appear that the mathematics of Lewin’s work is a language of scientific positivism, the emphasis on perceptual context and interpretation actually distances GMIT’s theory from scientific theory—at least the kind of “covering law” theory often cited in connection with scientific research. A music theory for communicating perceptions and intuitions locates music in experience and not in nature. » R. Satyendra, « An Informal Introduction to Some Formal Concepts from Lewin’s Transformational Thery », Journal of Music Theory, 48, pQuelle philosophie pour l’analyse transformationnelle ?

71 Se (dé)placer dans un orbifoldT2 = R/12Z x R/12ZT2 /S2Figure S9. (a) The space of ordered two-note chords of pitch-classes is a 2-torus. To identify points (x, y) and (y, x), we need to fold the torus along the AB diagonal. The resulting figure, shown in (b), is a triangle with two of its sides identified. This is a Möbius strip. To see why, cut figure (b) along the line CD and glue AC to CB. (To make this identification in Euclidean 3-space, you will need to turn over one of the pieces of paper.) The result is a square with opposite sides identified, as in Figure 2 of the main paper.Dmitri Tymoczko :« The Geometry of Musical Chords », Science, 313, 2006